TY - JOUR
T1 - Bifurcation and stability analyses for a coupled Brusselator model
AU - Yu, P.
AU - Gumel, A. B.
N1 - Funding Information:
The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.
Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.
PY - 2001/7/26
Y1 - 2001/7/26
N2 - This paper addresses the dynamic behaviour of a chemical oscillator arising from the series coupling of two Brusselators. Of particular interest is the study of the associated Hopf bifurcation and double-Hopf bifurcations. The motion of the oscillator may either be periodic (bifurcating from a Hopf-type critical point), or quasi-periodic (bifurcating from a compound critical point). Furthermore, bifurcation analysis reveals that the limit cycles associated with the first Brusselator are always stable, while that generated by the second Brusselator may be unstable if the parameter values are chosen far from the stability boundary. It is interesting to note that in the vicinity of the double-Hopf compound critical point, there exist periodic as well as quasi-periodic solutions. The quasi-periodic motion is stable for a small parameter region. A robust Gauss-Seidel like implicit finite-difference method (GS1) has been developed and used for the solution of the resulting initial-value problem (IVP). In addition to being of comparable accuracy (judging by the similarity of the profiles generated) with the fourth order Runge-Kutta method (RK4), the GS1 method will be seen to have better numerical stability property than RK4. Unlike the RK4, which fails when large time steps are used to integrate the IVP, extensive numerical simulations with appropriate initial data suggest that the GS1 method is unconditionally convergent. Moreover, it is more economical computationally.
AB - This paper addresses the dynamic behaviour of a chemical oscillator arising from the series coupling of two Brusselators. Of particular interest is the study of the associated Hopf bifurcation and double-Hopf bifurcations. The motion of the oscillator may either be periodic (bifurcating from a Hopf-type critical point), or quasi-periodic (bifurcating from a compound critical point). Furthermore, bifurcation analysis reveals that the limit cycles associated with the first Brusselator are always stable, while that generated by the second Brusselator may be unstable if the parameter values are chosen far from the stability boundary. It is interesting to note that in the vicinity of the double-Hopf compound critical point, there exist periodic as well as quasi-periodic solutions. The quasi-periodic motion is stable for a small parameter region. A robust Gauss-Seidel like implicit finite-difference method (GS1) has been developed and used for the solution of the resulting initial-value problem (IVP). In addition to being of comparable accuracy (judging by the similarity of the profiles generated) with the fourth order Runge-Kutta method (RK4), the GS1 method will be seen to have better numerical stability property than RK4. Unlike the RK4, which fails when large time steps are used to integrate the IVP, extensive numerical simulations with appropriate initial data suggest that the GS1 method is unconditionally convergent. Moreover, it is more economical computationally.
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U2 - 10.1006/jsvi.2000.3535
DO - 10.1006/jsvi.2000.3535
M3 - Article
AN - SCOPUS:0035954640
SN - 0022-460X
VL - 244
SP - 795
EP - 820
JO - Journal of Sound and Vibration
JF - Journal of Sound and Vibration
IS - 5
ER -